Mathematical Problems in Engineering

Volume 2019, Article ID 6139379, 18 pages

https://doi.org/10.1155/2019/6139379

## Temporal Performance Analysis of Bus Transportation Using Link Streams

^{1}Department of Electrical Engineering, Federal Institute of Education, Science and Technology of Santa Catarina (IFSC), Joinville 89220-618, Brazil^{2}Graduate Program in Electrical and Computer Engineering (CPGEI), Federal University of Technology-Paraná (UTFPR), Curitiba 80230-901, Brazil

Correspondence should be addressed to J. L. Curzel; rb.ude.csfi@lezruclj

Received 17 October 2018; Revised 31 December 2018; Accepted 17 January 2019; Published 11 February 2019

Academic Editor: Elena Zaitseva

Copyright © 2019 J. L. Curzel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Performance analysis of transport systems usually requires transfer of passengers between trains, cars, or buses, among others. This paper proposes a methodology for modeling and analysis of bus transportation using link streams. Link streams are particular cases of stream graphs whose cliques provide information about available time intervals for connecting buses. These cliques are obtained by algorithms of the literature with good scalability. They are used to quantify performance indicators as transfer time, bunching, congestion, and number of transferred passengers. The results are obtained for real-world data of a bus terminal in the city of Curitiba, Brazil. They reveal important issues regarding transfer delays and available capacity for transport. The proposed performance analysis can be used to support urban planners on planning and improving transport operation.

#### 1. Introduction

The operation of urban transport systems has been considered in the literature using different approaches [1–6]. These works model and simulate transport systems, particularly for planning schedules to improve system efficiency. The authors of [1, 2] developed a computational framework for modeling a transport system based on planning, station design, integration, and access and evaluated their effects on system performance. In [3] a parallel operation management system has been developed to detect the number of passengers at stations as well as traffic flow and queuing length of vehicles in private lanes. This system provides information to deal with transportation management by improving and optimizing emergency situations or rescheduling vehicles based on conditions detected in videos of traffic. In [4–6] simulation and intelligent support based on artificial intelligence techniques are used to help control center operators to take strategic decisions about a fleet of buses in an urban network.

Recently, transportation companies have invested in technologies for online supervision of operations and resources [7, 8]. For instance, a two-way communication in [7] between bus drivers and operators allows solutions using negotiation-based strategies. In [8] a global positioning system (GPS) is integrated with wireless communication to track vehicles in real time. All these approaches utilize information and communication technology (ICT) to predict, simulate, and control different situations in transport system. In this sense, the available online information can be used for planning, controlling, and coordinating transit of buses in a reactive manner. However, formal methodologies are still necessary to analyze and detect issues in advance.

This paper aims to analyze multiple connections in a bus terminal. By means of an innovative performance analysis method based on link streams [9], we model a transport system of multiple bus line connections in a bus terminal (or hub). Link streams are particular cases of stream graphs which are graphs whose nodes and edges (links) have a time to live. A link stream is a stream graph in which only edges have a time to live; i.e., nodes are permanent. A detailed introduction of stream graphs and link streams can be found in [10]. Links streams have been used in many applications. For instance, nodes and edges in a graph may represent individuals connected by a phone call which persists for a given period of time. The authors of [11] use link streams to capture dynamics of contacts between individuals. They consider activities as link streams to predict the number of links that occur during a given period of time and show that a combination of structural and temporal characteristics of the link stream leads to performance improvements. In [12] a trace of real-world interactions between individuals captured with RFID sensors technology was collected in a high school in France during approximately 8 days of 2012. The study of links created and destroyed during these days has revealed new patterns of human interaction in a different time scale. Another important issue in link streams is computing cliques. A clique is a set of nodes such that any 2-combination of these nodes is connected by an edge. A first algorithm to compute maximal cliques in a link stream was proposed in [9] and implemented in [13]. Recently, it was improved in [14, 15] by adapting the Bron-Kerbosch algorithm. Moreover, the authors developed an algorithm that detects maximal time intervals during which interactions occur. This notion was initially introduced by [9] using instantaneous link streams and extended to link streams with duration by [16].

Our main contribution is to define a set of procedures to represent arrivals and departures of buses in a link stream representation whose cliques provide the necessary information to compute performance indicators as transfer times, bunching, congestion, and number of passengers transferred between buses. To the best of our knowledge, this is the first time link streams are used to model and analyze transport systems.

The paper is organized as follows. Section 2 presents the background on link streams and cliques. Section 3 shows how link streams and cliques model a bus transportation system and defines indicators for the performance analysis. Section 4 presents the case study of Curitiba’s public transportation with conclusions in Section 5.

#### 2. Link Streams

A simple undirected graph is an ordered pair with a set of nodes and a set of links with for , and means a set of unordered pairs of nodes). In this case, nodes and are linked together in .

Stream graphs and link streams are graphs with time information. A comprehensive study on stream graphs and link streams can be found in [10]. A stream graph is a tuple such that is a time interval, is a set of temporal nodes, and is a set of links. If then and . This means that nodes and and a link exist at time . A stream graph is then a graph whose nodes and links exist at a given time . If then all nodes exist for all times in , and is a link stream.

Links streams are used to model interactions between individuals or objects over time as meetings in social networks or email exchanges [12]. In this paper, we use link streams for modeling transport systems because nodes (bus stops) do exist during the whole period of time considered for analysis. However, links between nodes only occur at given time intervals during which passengers are transferred.

In a simple undirected graph , a clique of is a set of nodes which are connected to each other; i.e., for all , there exists a link . A clique is maximal if there is no other clique such that .

A similar definition is used for a stream graph . According to [10], a clique of is a subset of such that all pairs of nodes involved in are linked in . Because nodes in stream graphs are not present all the time in general, it is necessary to distinguish compact cliques which are subsets of whose nodes are linked all together during a given time interval. This is the case for link streams which are compact stream graphs [10] as all nodes are present within . Therefore, a (compact) clique in a link stream is given by a time interval and a subset of nodes which are linked all together during this time interval. Similarly, a clique is maximal if there is no other clique such that . Figure 1 shows examples of cliques in a link stream .